Bloch Waves in Bubbly Crystal Near the First Band Gap: A High-Frequency Homogenization Approach

다국어 초록 (Multilingual Abstract)

<P>This paper is concerned with the high-frequency homogenization of bubbly phononic crystals. It is a follow-up of the work [H. Ammari et al., <italic toggle='yes'>J. Differential Equations</I>, 263 (2017), pp. 5610--5629], which shows the existence of a subwavelength band gap. This phenomena can be explained by the periodic inference of cell resonance which is due to the high contrast in both the density and bulk modulus between the bubbles and the surrounding medium. In this paper, we prove that the first Bloch eigenvalue achieves its maximum at the corner of the Brillouin zone. Moreover, by computing the asymptotic of the Bloch eigenfunctions in the periodic structure near that critical frequency, we demonstrate that these eigenfunctions can be decomposed into two parts: one part is slowly varying and satisfies a homogenized equation, while the other is periodic across each elementary crystal cell and is varying. They rigorously justify, in the nondilute case, the observed superfocusing of acoustic waves in bubbly crystals near and below the maximum of the first Bloch eigenvalue and confirm the band gap opening near and above this critical frequency.</P>